Volume 2009, Article ID 841045,11pages doi:10.1155/2009/841045
Research Article
One-Time Key Based Phase Scrambling for Phase-Only
Correlation between Visually Protected Images
Izumi Ito (EURASIP Member) and Hitoshi Kiya (EURASIP Member)
Graduate School of System Design, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino-shi, Tokyo 191-0065, Japan
Correspondence should be addressed to Hitoshi Kiya,[email protected]
Received 24 March 2009; Revised 18 August 2009; Accepted 23 October 2009
Recommended by Sen-Ching Samson Cheung
One-time key based phase scrambling is proposed for privacy-protected image matching. The image matching is performed using invisible templates that are protected by phase scrambling. A key for phase scrambling is not required for the image matching, and the key can even be discarded after scrambling if the template does not need to be reconstructed. Theoretical analyses are presented to provide guidelines for designing key parameters that affect the visual effect and image matching. Experimental results demonstrate the effectiveness and appropriateness of the proposed method.
Copyright © 2009 I. Ito and H. Kiya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
A number of approaches to image matching, such as correlation in the space domain and using features as typified by corners and edges, have been investigated. Phase-only correlation (POC) is a frequency domain approach to image matching. Phase-only correlation with discrete Fourier transform (DFT) was first proposed by Kuglin and Hines [1]. The translation between signals and the direct measure of the degree of signal congruence can be simultaneously estimated by POC based on the Fourier shift property. In addition, the rotation and scaling can be estimated using the magnitude of DFT coefficients that are mapped into log-polar coordinates [2]. High-accuracy estimation by POC has been developed [3–5].
Generally, image matching using POC requires visual protection of templates in order to secure privacy [6, 7]. Typically, encryption is used for the protection of signals [8]. However, decryption is required before image matching using POC. In order to address these problems, we previously proposed a method for image matching that used synchro-nized phase scrambling [9, 10]. The previously proposed method enables direct image matching between protected images. However, a key is required for both scrambling and image matching, and the key must be kept secret from attackers.
In one-time key based phase scrambling, a key is used once for scrambling but is not required for image matching. Moreover, after scrambling, the key can be discarded if the template does not need to be reconstructed. Under the limited two-member set, the effect of scrambling on POC values and the visual effect are analyzed theoretically, and these analyses provide a guideline for designing key parameters. A key for which the parameters have been chosen appropriately enables keyless image matching to be performed by phase scrambling. Finally, experimental results demonstrate the effectiveness and appropriateness of the proposed scrambling.
2. Preliminary
The goal of the proposed image matching method is described in this section. Phase-only correlation and phase scrambling for POC, which are elements of the proposed method, are then explained. In the present paper, for the sake of brevity, the one-dimensional case is considered. LetC,R, andZdenote the sets of complex, real, and integer numbers, respectively.
2.1. Goal of the Proposed Image Matching. Phase scrambling
A template
DFT
DFT coefficients Inverse
DFT
Phase
scrambling Key
Phase-scrambled DFT coefficients
Inverse DFT
Phase-scrambled image (visual protection)
Figure1: Phase scrambling. The phase scrambling is accomplished in the frequency domain. The inverse DFT of the DFT coefficients reveals the information of the template. Conversely, the inverse DFT of the scrambled DFT coefficients does not reveal the information of the template.
the frequency domain. The DFT coefficients of a signal are scrambled by multiplying the coefficients with random phase terms. The inverse DFT of the scrambled DFT coefficients does not reveal the original information of the image, whereas the inverse DFT of the DFT coefficients reveals the information of the original image.
The proposed image matching uses phase scrambling in order to protect the original information of templates visually in case there is leakage of the template. Image matching between the phase-scrambled template and the query can be performed by POC even if the sensed image that is used as a query has translation, rotation, and scaling for the corresponding template.
The difference between the previously proposed image matching and the image matching proposed herein is shown in Figure 2. In the previous method, the key used for the phase scrambling of templates must be saved securely and reused in the image matching by POC. On the other hand, in the proposed method, the key used for phase scrambling of templates is not required for image matching and does not need to be saved. Independent keys are used for all templates and the key is a one-use key.
2.2. POC
2.2.1. Translation. Let Gi(k), k = 0, 1,. . .,N − 1, i ∈ Z be the N-point DFT coefficients of N-point signal,
gi(n) ∈ R,n = 0, 1,. . .,N −1. The phase term φGi(k) is
defined as
φGi(k)= Gi(k)
|Gi(k)|, (1)
where |Gi(k)| denotes the absolute value of Gi(k). If
|Gi(k)| =0, thenφGi(k) is replaced by 0.
Letg2(n) be the shifted signal ofg1(n). The normalized cross spectrum,Rφ(k), betweeng1(n) andg2(n) is defined in
terms of their corresponding phase termsφG1(k) andφG2(k)
as follows:
Rφ(k)=φ∗G1(k)·φG2(k), (2)
whereφ∗G1(k) denotes the complex conjugate ofφG1(k). The
POC function,rφ(n), is defined by the inverse DFT ofRφ(k)
as
rφ(n)=
1
N
N−1
k=0
Rφ(k)WN−nk, (3)
whereWNdenotes exp(−j2π/N) andjdenotes
√
−1 [1]. The translation between two signals is estimated by the location of the peak ofrφ(n) in (3). In addition, the value of the peak
is used as a measure of the signal congruence.
2.2.2. Rotation and Scaling. Rotation and scaling between
two images are estimated by POC using the magnitude of DFT coefficients that are mapped into log-polar coordinates [2]. Log-polar transform reduces the rotation angle and scale factor in the Cartesian coordinates to horizontal and vertical translation in log-polar coordinates.
2.3. Phase Scrambling for POC
2.3.1. Phase Scrambling and Phase-Scrambled Signal. Phase
scrambling for POC is performed in the frequency domain. The phase-scrambled DFT coefficients,Gi(k), are defined as
Gi(k)=Gi(k)·ejθαi(k), (4)
where θαi(k) denotes a key sequence and αi denotes an
identifier for the key sequence. Phase scrambling affects only the phase term, that is, the scrambled phase term,φGi(k), is
given as
φGi(k)=φGi·e
jθαi(k). (5)
The phase-scrambled signal, gi(n), is defined by the
inverse DFT ofGi(k) as
gi(n)=
1
N
N−1
k=0
Gi(k)WN−nk. (6)
The phase-scrambled image is a two-dimensional expression of the phase-scrambled signal.
A key sequence is constructed from a set ofMmembers,
Template no.i A query image
DFT DFT
DFT coefficients DFT coefficients
Phase scrambling Phase
scrambling A key
Phase-scrambled DFT coefficients
Phase-scrambled DFT coefficients
POC
(a) previous image matching
Template no.i
A query image DFT
DFT coefficients
DFT Phase
scrambling Keyi
Phase-scrambled
DFT coefficients DFT coefficients
POC
(b) proposed image matching
Figure2: Difference between the previously proposed image matching and the image matching proposed herein. (a) Previously proposed image matching. A key is required for scrambling both the templates and the query images. (b) Proposed image matching. A key is only used for scrambling the templates. All of the templates can be scrambled by independent keys.
the same as that of the DFT coefficients. That is, anM-ary key sequence,θαi(k), is expressed with a set,UxM1, as
θαi(k)∈U
M x1, ∀k,
UxM1 = {x1,x2,. . .,xM},
(7)
where, for convenience, the superscript and subscript of the term UM
x1 denote the number of members and the
first member, respectively. A key sequence can be generated using a cryptographically secure pseudorandom number generator, in which the random numbers are related to each member ofUM
xi specified by the users. AsM increases, the
key space increases, whereas the determination of members becomes more complicated.
2.3.2. Image Matching under Synchronized Phase Scrambling.
The phase scrambled signal can be used directly for image matching using POC.
Letg1(n) and g2(n) be a template and a query, respec-tively. According to (4),g1(n) is scrambled byθα1(k) and then
stored in the form of phase-scrambled DFT coefficients in a system. When the system is queried with respect tog2(n),
g2(n) is scrambled byθα2(k), and the POC betweenG1(k) and
G2(k) is then performed. In other words,G1(k) andG2(k) are given as
G1(k)=G1(k)·ejθα1(k),
G2(k)=G2(k)·ejθα2(k).
(8)
The normalized cross spectrumRφ(k) is calculated as
Rφ(k)=φ∗G1(k)·φG2(k). (9)
The POC functionrφ(n) is then obtained as
rφ(n)= 1
N
N−1
k=0
Rφ(k)WN−nk. (10)
When the key sequence for the template and that for the query are the same, the POC under phase scrambling and the POC between nonscrambled signals are identical. Namely, if for allk,θα1(k)=θα2(k), then by substituting (5) into (9), we
obtain the following:
Rφ(k)=φ∗G1(k)e
−jθα1(k)·φG 2(k)e
jθα1(k)=Rφ(k). (11)
From (3), (10), and (11), we therefore obtain
0 π
δ
(a)U02= {0,π}
0 π/2
−π/2
δ
(b)Uπ/22= {π/2,−π/2}
0 π/3
−π/3
δ
(c)Uπ/23= {π/3,−π/3}
0 π/6
−π/6
δ
(d)Uπ/26= {π/6,−π/6} Figure3: Difference of phases.
Thus, a key sequence is used for both the template and the query, and these two key sequences are synchronized. In the proposed method, a key sequence is used for a template only.
3. Proposed Image Matching and Visual
Information Protection
One-time key based phase scrambling is proposed and analyzed statistically. In the present paper, the set for key sequences is limited to a two-member set.
3.1. One-Time Key Based Phase Scrambling for POC. A
binary key sequence is used only once for scrambling a template and is not required for image matching.
The binary key sequence is constructed from a set of only two members,x1andx2, that is,
θα1(k)∈U
2
x1= {x1,x2}. (13)
For anN-point signal, a binary key sequence of lengthNis generated using a cryptographically secure pseudo random number generator, the output of which belongs to eitherx1
orx2. Letqx1be the occurrence probability ofx1, and letδbe
the difference of phases,x1−x2, examples of which are shown inFigure 3.
Letg1(n) and g2(n) be a template and a query, respec-tively. In a system, g1(n) is scrambled by θα1(k) ∈ Ux21
according to (4) and is stored in the form ofG1(k). When the system is queried with respect tog2(n), its DFT coefficients,
G2(k), are multiplied by a constant, that is,
G1(k)=G1(k)·ejθα1(k), (14)
G2(k)=G2(k)·ejx1(k). (15)
The POC betweenG1(k) andG2(k) is calculated according to (9) and (10).
Although (12) is not satisfied under one-time key based phase scrambling, the peak value ofrφ(n), which is used as a
measure of signal congruence, is estimated fromrφ(n) using
parameters that are explained in the following section.
3.2. Effect of One-Time Key Based Phase Scrambling. The
effect on the peak value of POC is considered in terms of the average value of POC under the proposed scrambling.
From (5), (9), (14) and (15),Rφ(k) is expressed as
Rφ(k)=φ∗G1(k)·φG2(k)
=φ∗G1(k)e
−jθα1(k)·φG 2(k)e
jx1
=Rφ(k)ej(x1−θα1(k)).
(16)
Here, we assume that the key sequence is a single value, that is, if for allk,θαi(k)=xC, then
Rφ(k)=Rφ(k)ej(x1−xC) (17)
and since ej(x1−xC) is a constant, from (10), r
φ(n) can be
obtained as
rφ(n)=ej(x1−xC)rφ(n). (18)
Under the above assumption, ifxC=x1, then
rφ(n)=rφ(n), (19)
and ifxC=x2, then
rφ(n)=ej(x1−x2)rφ(n). (20)
From (19) and (20), the average value of rφ(n), that is,
rφave(n), is defined in terms ofqx1as follows:
rφave(n)=qx1·rφ(n) +
1−qx1
·ejδ·r
φ(n). (21)
Since rφave(n) is a complex number, the real and imaginary
parts ofrφave(n) can be expressed as
Re rφave(n)
=qx1rφ(n) +
1−qx1
·cos(δ)·rφ(n),
Im rφave(n)
=1−qx1
·sin(δ)·rφ(n),
(22)
where Re[·] and Im[·] denote the operations to obtain the real and imaginary parts, respectively, of signals, and their peak values can be given as
max
n
Re r
φave(n)
=qx1p+
1−qx1
·cos(δ)·p, (23)
max
n
Im r
φave(n)
=1−qx1
·sin(δ)·p, (24)
wherepdenotes the peak value ofrφ(n), which is referred to
The peak value of POC under one-time key based phase scrambling is approximated by the average value of POC under one-time key based phase scrambling. That is,
max
n
Re r
φ(n)
≈max
n
Re r
φave(n)
,
max
n
Im r
φ(n)
≈max
n
Im r
φave(n)
.
(25)
Based on the above consideration, we observe the following.
Adjustment of the Peak Value. The peak value ofrφ(n) can
be controlled using the occurrence probability,qx1, and the
difference of phases,δ, as parameters. If queries are expected to contain noise, which causes a lower peak value, the peak value can be adjusted using parametersqx1andδto be higher
within (23) and (24).
Invalid Parameters for the Proposed Scrambling. There is a
set of invalid parameters for one-time key based phase scrambling. From (24), we obtain δ = π + 2πl, l ∈ Z such that, for qx1=/0, maxn (|Im[rφave(n)]|) = 0. From
(23), we obtain qx1 = 0.5 such that, for δ = π + 2πl,
max
n (|Re[rφave(n)]|)=0. The set of invalid parameters is thus
given as
qx1,δ
=(0.5,π+ 2πl), l∈Z. (26)
Estimation of the Original Peak Value. The original peak
value, p, can be estimated fromrφ(n) using parametersqx1
andδ, and thereby the effect of scrambling can be practically avoided. From (23) and (24), the original peak value,p, can be estimated from the observed peak value using the above parameters as follows:
p≈ maxn
Re r
φ(n)
qx1+
1−qx1
·cos(δ),
p≈maxn
Im r
φ(n)
1−qx1
·sin(δ).
(27)
3.3. Effect on Visual Information. We consider the error
energy between the phase scrambled signal and the non-scrambled signal, in which the larger error energy provides greater visual protection.
Letgi(n) be the scrambled signal of the original signal,
gi(n), obtained using the key sequenceθαi(k), that is,
gi(n)=
1
N
N−1
k=0
Gi(k)ejθαi(k)WN−nk. (28)
Here, we assume that the key sequence is a single value, that is, if for allk,θαi(k)=xC, then
gi(n)= 1
N
N−1
k=0
Gi(k)ejxCWN−nk
=ejxC1 N
N−1
k=0
Gi(k)WN−nk
=ejxCg
i(n).
(29)
Under the above assumption, since gi(n) is a complex
number, the real and imaginary parts ofgi(n) are expressed
accordingly as
Regi(n)
=cos(xC)·gi(n), (30)
Imgi(n)
=sin(xC)·gi(n). (31)
The real part of the error energy,Er, between the original
signal and the real part of the phase scrambled signal is defined as
Er= N−1
n=0
gi(n)−Re
gi(n)2. (32)
Under the above assumption, from (30), (32) is expressed as
Er= N−1
n=0
|1−cos(xC)|2gi(n)2
= |1−cos(xC)|2·Cgi,
(33)
where
Cgi=
N−1
n=0
gi(n)2
. (34)
IfxC=x1, then
Er= |1−cos(x1)|2·Cgi (35)
and ifxC=x2, then
Er= |1−cos(x2)|2·Cgi. (36)
The average,Er, of the real part of the error energy,Er, is
defined in terms of the occurrence probability,qx1, as
Er=qx1|1−cos(x1)|
2· Cgi+
1−qx1
|1−cos(x2)|2· Cgi.
(37)
The average of the imaginary part of the error energy is defined in a similar manner. The imaginary part of the error energy,Ei, between the original signal and the imaginary part
of phase scrambled signal is defined as
Ei= N−1
n=0
gi(n)−Im
(a) template (b) query
Figure4: Template and query. (a) Template: 512×512, 8 bits/pixel. (b) The query is generated by translation from (a) by 20 pixels in the horizontal and vertical directions.
The average,Ei, ofEiis defined withqx1as
Ei=qx1|1−sin(x1)|
2· Cgi+
1−qx1
|1−sin(x2)|2· Cgi.
(39)
The visual effect of phase-scrambled signals can be estimated from (37) and (39), as will be shown inSection 4.2. The parameters for key sequence are selected based on both the average error energy and the effect on the peak value discussed inSection 3.2.
3.4. Security Consideration. A key sequence based on a
user-specified key is generated using a cryptographically secure pseudorandom number generator. A key sequence is used only once for scrambling a template. If the scrambled template does not need to be reconstructed in a system, the key sequence can be discarded after scrambling, and, consequently, its protection is not required. In the following, we focus on the known-plaintext attack and the ciphertext-only attack.
3.4.1. Known-Plaintext Attack. This model assumes that an
attacker has samples of both a plaintext and its ciphertext and uses them to reveal further secret information, such as secret keys and code books.
In a system using the proposed scrambling, each template is scrambled by an independent key sequence. Accordingly, even if an attacker obtains sets of a plaintext and its ciphertext, the other key sequences cannot be inferred by these sets.
3.4.2. Ciphertext-Only Attack. This model assumes that an
attacker accesses only a ciphertext.
In this model, there are two approaches by which to deduce the original template:
(1) using a local image possessed by an attacker. (2) guessing a key sequence.
With respect to the former approach, although the original template can be inferred using the signal congruence from the POC between a scrambled template and a local image
possessed by an attacker, or through other methods using such a local image, even if an attack is successful, the attacker has already had access to a closely related image; the contents of which are already known to the attacker prior to the attack. Namely, the situation is considered as a case in which the information has been already leaked. The aim of the proposed scrambling is to prevent the information leakage from a template itself. The inference by using a local image is therefore excluded from the scope of the protection of the proposed scrambling.
With respect to the latter approach, theoretically, even if in the case of a brute force attack, as in the case of the one-time pad, which is unconditionally secure and theoretically unbreakable, it is impossible to confirm whether an inferred key sequence is correct. Since not a binary bit but a coherent unit of phase is changed, the scrambled template can be practically inferred by brute force attack although complete restoration is impossible for the above-described reason. Consequently, the proposed scrambling protects the visual information of the original template within the scope of key space and completely protects the original template. If the size of a template isN×N, the key space is 2N×N. We assume
the size of a template to be adequate.
When a system requires much higher security, the proposed scrambling can be combined with other crypto-graphic techniques. Even if decryption is required for image matching, the effect of the proposed scrambling remains valid.
4. Simulations
4.1. POC under the Proposed Scrambling. We performed
POC between two images under one-time key based phase scrambling to show that both translation and signal congru-ence can be estimated even under scrambling.
In the following simulations, we used a 512×512 section of a 1, 024×1, 024, 8-bits/pixel image called “pentagon”, as a template and a query, as shown inFigure 4. The template and query were translated by 20 pixels in the horizontal and vertical directions. The template was scrambled by a key sequence,θαi(k1,k2), according to (4). After the query was
processed according to (15), POC was performed between the template and the query. The translation and signal congruence were then estimated from the observed peak of the POC surface.
4.1.1. Without Noise. First, we evaluated the peak value
by controlling the parameters for key sequences. The key sequence θαi(k1,k2) was generated using the sets of
{π/2,−π/2}, that is, δ = π with qπ/2 = 0.5 to 0.95,
and{a,−a}, a = π/8 to π/2 with qa = 0.5, respectively.
The original peak value, p, which is the peak value of the POC between the non-processed template and query, was 0.8264.
The real parts of POC surfaces for the cases in which
θαi(k1,k2) ∈ {π/2,−π/2}withqπ/2 = 0.7, 0.6, and 0.5, are
0
20
40
n1
0 20
40
n2 0 0.1 0.2 0.3
Re
[
rφ
(
n1
,
n2
)]
Peak value 0.3291
(a) pπ/2=0.7
0
20
40
n1
0 20
40
n2 0 0.1 0.2 0.3
Re
[
rφ
(
n1
,
n2
)] Peak value
0.1675
(b) pπ/2=0.6
0
20
40
n1
0 20
40 n2 0 0.1 0.2 0.3
Re
[
rφ
(
n1
,
n2
)] Peak value
0.0062
(c)pπ/2=0.5
Figure5: Real part of the POC surface without noise.θαi(k1,k2)∈ {π/2,−π/2}, (δ=π). The peak value decreases according to (23) asqπ/2
approaches 0.5.
0
20
40
n1
0 20
40
n2
−0.05
0 0.05 0.1 0.15
Re
[
rφ
(
n1
,
n2
)]
Peak value 0.1198
(a) real part
0
20
40
n1
0 20
40
n2 0 0.1 0.2 0.3
Im
[
rφ
(
n1
,
n2
)]
Peak value 0.2926
(b) imaginary part
1 256 512 Horizontal translation
1 256 512
V
er
tical
tr
anslation
(a)θαi(k1,k2)∈ {π/2,−π/2},qπ/2=0.5
1 256 512
Horizontal translation 1
256 512
V
er
tical
tr
anslation
(b)θαi(k1,k2)∈ {π/2,−π/2},qπ/2=0.7
1 256 512
Horizontal translation 1
256 512
V
er
tical
tr
anslation
(c)θαi(k1,k2)∈ {π/3,−π/3},qπ/3=0.5
Figure 7: Translation estimation without noise using a total of 1,000 different key sequences. The “×” symbols denote the estimated translation, which is the peak location on the POC surface. The “∇” symbols denote the degeneracy of the “×” plots. The translation can be estimated correctly for all but the set of invalid parameters.
peak value was confirmed to decrease according to (23) as
qπ/2 approaches 0.5 for δ = π, and, for the case in which δ=πandqπ/2=0.5, that is, the set of invalid parameters, a
distinct peak did not appear on the POC surface.
The real and imaginary parts are shown in Figures6(a)
and 6(b), respectively, of the POC surface for the case in whichθαi(k1,k2) ∈ {3π/8,−3π/8}, that is,δ = 3π/4, with q3π/8 =0.5. A peak appears in both the real and imaginary
parts of the POC surface.
The error between the observed and calculated peak values are presented in Tables1 and2. The observed peak values in the tables denote the average peak values observed on the POC surface for a total of 20 different key sequences. The peak value of POC under the proposed scrambling can be adjusted by controlling the parameters according to (23) and (24), which also means that the original peak value
can be estimated from the observed peak value using the parameters according to (27).
Next, we investigated the peak location on the POC surface. A total of 1,000 different key sequences, which were generated using the sets of {π/2,−π/2} with qπ/2 = 0.5, {π/3,−π/3}withqπ/3 = 0.5, and{π/2,−π/2}withqπ/2 =
0.7, were considered.
The estimated locations for different key sequences are shown in Figure 7. For the case in which θαi(k1,k2) ∈
{π/2,−π/2} with qπ/2 = 0.5, that is, the set of invalid
parameters, the translation was not estimated correctly, as shown in Figure 7(a). For the cases in whichθαi(k1,k2) ∈
{π/2,−π/2}withqπ/2 = 0.7 andθαi(k1,k2)∈ {π/3,−π/3}
withqπ/3 = 0.5, the translation was estimated correctly, as
0
20
40
n1
0 20
40 n2 0 0.05 0.1
Re
[
rφ
(
n1
,
n2
)]
Peak value 0.0982
pπ/2=0.7
40
20
0 n2
0
20
40
n1
0 0.05 0.1
Re
[
rφ
(
n1
,
n2
)]
pπ/2=0.5 (a) with additive noise
0
20
40
n1
0 20
40 n2 0 0.04 0.08
Re
[
rφ
(
n1
,
n2
)]
pπ/2=0.7
40
20
0 n2
0
20
40
n1
0 0.04 0.08
Re
[
rφ
(
n1
,
n2
)]
Peak value 0.0714
pπ/2=0.5 (b) with JPEG quantization noise
Figure8: Real part of POC surface with noise.θαi(k1,k2)∈ {π/2,−π/2}. (a) The query is corrupted by additive noise (Gaussian random
numbers with zero mean and a standard deviation of 25). (b) The query is corrupted by JPEG quantization noise (Q-factor=200).
Table1: Error between the observed and calculated peak values of the real part of the POC surface.θαi(k1,k2)∈ {π/2,−π/2}, that is,δ=π,
withqπ/2=0.55 to 0.95.
qπ/2 Observed peak value Calculated peak value Error
0.55 0.0829 0.0826 0.0003
0.60 0.1653 0.1652 −0.0001
0.65 0.2482 0.2478 0.0004
0.70 0.3307 0.3304 0.0003
0.75 0.4136 0.4130 0.0006
0.80 0.4954 0.4956 −0.0002
0.85 0.5785 0.5782 0.0003
0.90 0.6614 0.6608 0.0006
qπ/2=0.9 qπ/2=0.8 qπ/2=0.7 qπ/2=0.6 qπ/2=0.5 (a) real part
qπ/2=0.9 qπ/2=0.8 qπ/2=0.7 qπ/2=0.6 qπ/2=0.5 (b) imaginary part
Figure 9: Real and imaginary parts of phase-scrambled images. Image “Lena” (512×512, 8 bits/pixel) is scrambled byθαi(k1,k2) ∈
{π/2,−π/2}, that is,δ=πwith occurrence probabilityqπ/2=0.5, 0.6, 0.7, 0.8, and 0.9.
0 π/8 π/4 3π/8 π/2
a(rad)
qa=0.9
qa=0.8
qa=0.7
qa=0.6
qa=0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Er
/C
gi
(a) real part
0 π/8 π/4 3π/8 π/2
a(rad)
qa=0.9
qa=0.8
qa=0.7
qa=0.6
qa=0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Ei /Cg
i
(b) imaginary part
Figure10: Average error energy overCgi. Image “Lena” (512×512, 8 bits/pixel) is scrambled byθαi(k1,k2)∈ {π/2,−π/2}, that is,δ=π
with occurrence probabilityqπ/2=0.5, 0.6, 0.7, 0.8, and 0.9.
Table2: Error between the observed and calculated peak values of the real and imaginary parts of the POC surface.θαi(k1,k2)∈ {a,−a},
that is,δ=2a, withqa=0.5.
a Part Observed peak value Calculated peak value Error
π/2 Real 0.0064 0.0000 0.0064
π/2 Imaginary 0.0064 0.0000 0.0064
3π/8 Real 0.1212 0.1210 0.0002
3π/8 Imaginary 0.2924 0.2922 0.0002
π/4 Real 0.4132 0.4132 0.0000
π/4 Imaginary 0.4131 0.4132 −0.0001
π/8 Real 0.7055 0.7054 0.0001
has been demonstrated to be possible for all but the set of invalid parameters.
4.1.2. With Noise. We evaluated the noise version of POC
under the proposed scrambling. The key sequence was generated by the set of{π/2,−π/2}withqπ/2=0.7 and 0.5.
The POC surface between the template and the query that was corrupted by additive noise is shown inFigure 8(a). The noise consists of Gaussian random numbers with zero mean and a standard deviation of 25. Under this condition, the original peak value was 0.2433. The POC surface between the template and the query that was corrupted by JPEG quantization noise is shown inFigure 8(b). TheQ-factor was 200.( 8×8 DCT coefficients are divided by the quantization step Q(k1,k2), which is generated by QT(k1,k2)· QF/50,
whereQT(k1,k2), andQFdenotes a predefined quantization
table and aQ-factor, resp.). Under this condition, the original peak value was 0.1779. Whenqπ/2=0.7, a peak that expresses
the translation appeared on the POC surface at the location (20, 20) in both conditions and the peak values were 0.0982 and 0.0714, respectively. We confirmed that even when the query is corrupted by noise, the original peak value under the condition can be estimated from the observed peak value on the POC surface under the proposed scrambling.
4.2. Visual Effect and the Average Error Energy. We compared
the visual effect of a phase-scrambled image with the average error energy in order to demonstrate the validity of the discussion inSection 3.3. We used a 512×512, 8 bits/pixel image called “Lena”. The key sequence was generated from the set of{π/2,−π/2}, withqπ/2=0.5, 0.6, 0.7, 0.8, and 0.9.
The average error energy was calculated according to (37) and (39) using the parameters{a,−a}, that is,δ = 2a, for
a=0 toπ/2 andqπ/2=0.5, 0.6, 0.7, 0.8, and 0.9.
The real and imaginary parts of the phase-scrambled images are shown in Figures 9(a) and 9(b), respectively. The real part of the phase-scrambled images is invisible and shows almost no variation, whereas the imaginary part of the phase-scrambled images is degraded as the occurrence probability approaches 0.5. The real and imaginary parts of the average error energy calculated from (37) and (39) are shown in Figures 10(a)and10(b), respectively. When a = π/2, the average of the real part of the error energy is constant regardless of the occurrence probability, whereas the average of the imaginary part of the error energy increases as the occurrence probability approaches 0.5. The average error energy is confirmed to coincide with the visual effect of the phase-scrambled image.
If templates are required to be invisible, the parameters should be set as close to the set of invalid parameters as possible.
5. Conclusion
We have proposed one-time key based phase scrambling for image matching. Protecting the original information of the template visually for privacy and security, the proposed method enables keyless image matching. The occurrence probability of a member and the difference of phases were
discussed for the one-time key based phase scrambling. These parameters can control the effect of visual protection and the peak value of POC. The effectiveness of visual protection has been demonstrated by the error energy between the phase-scrambled image and the original image. The peak value of one-time key based phase scrambling for the case of the two-member set has been explained theoretically. Experimental results revealed the effectiveness and appropriateness of the proposed method. In the future, a theoretical explanation for the case of a multiple-member set and the degree of robustness against attacks will be considered.
Acknowledgment
This study was supported in part by a Grant-in-Aid for Scientific Research C, no. 20560361, from the Japan Society for the Promotion of Science (JSPS).
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